Missing step in engineering problem

I'm reading over a section in my statics text that describes an analysis of a cable with a distributed load, but it's been a long time since I've done this stuff, and I'm stuck on just one step. :grumpy:

That just comes from balancing the forces in one direction on a Δx long segment of the cable. So far so good. But then the text says "dividing this equation by Δx and taking the limit as Δx→0, Δθ→0 and ΔT→0, we obtain:" and then gives this:
d(T∙cos(θ))/dx = 0

Now, I'm trying to see how they got that, and failing. Likewise, for the forces in the y direction, they go from:
-T∙sin(θ) - ω(x)∙Δx + (T + ΔT)∙sin(θ + Δθ) = 0

To:
d(T∙sin(θ))/dx - ω(x) = 0

Now, the ω(x) component I have no problem with, but again, how do they go from "[-A∙f(B) + (A + ΔA)∙f(B + ΔB)] / Δx" to "d(A∙f(B))/dx" simply by taking the limits as the Δ's go to 0?